Percolation thresholds and epidemic spreading on small-world networks: Exact results and critical dynamics

Percolation theory provides a powerful framework for understanding the emergence of large-scale epidemic outbreaks on complex networks. Building on the seminal work of Moore and Newman (2000), we revisit and extend percolation-based models of disease spread on small-world networks that incorporate both local connections and long-range shortcuts. Specifically, we resolve two previously unresolved cases by deriving exact analytical expressions for the bond percolation threshold at local connectivity K=3, and for the hybrid site-bond percolation threshold at K=2. These results advance the theoretical foundations of epidemic modelling on structured networks. Complementing these theoretical results, we conduct extensive numerical simulations to characterize critical behaviour of both pure bond and hybrid site-bond percolation processes. We find that percolation thresholds are highly sensitive to shortcut density and local connectivity. Notably, the average size of finite percolating clusters exhibits a pronounced peak at criticality, offering a reliable early-warning signal for epidemic onset. Furthermore, we observe a clear transition in cluster size distributions — from exponential decay to heavy-tailed forms — as the system approaches the percolation threshold, culminating in the emergence of a giant component indicative of a large-scale epidemic. Temporal simulations further reveal abrupt epidemic transitions as control parameters cross critical thresholds, in agreement with our percolation-based predictions. Collectively, our results establish the percolation framework as a powerful tool for capturing both structural and dynamical features of epidemic spreading on small-world networks.